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Fast Cross-Polytope Locality-Sensitive Hashing

Authors: Christopher Kennedy and Rachel Ward

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We provide a variant of cross-polytope locality sensitive hashing with respect to angular distance which is provably optimal in asymptotic sensitivity and enjoys \mathcal{O}(d \ln d ) hash computation time. Building on a recent result in (Andoni, Indyk, Laarhoven, Razenshteyn '15), we show that optimal asymptotic sensitivity for cross-polytope LSH is retained even when the dense Gaussian matrix is replaced by a fast Johnson-Lindenstrauss transform followed by discrete pseudo-rotation, reducing the hash computation time from \mathcal{O}(d^2) to \mathcal{O}(d \ln d ). Moreover, our scheme achieves the optimal rate of convergence for sensitivity. By incorporating a low-randomness Johnson-Lindenstrauss transform, our scheme can be modified to require only \mathcal{O}(\ln^9(d)) random bits.

Cite as

Christopher Kennedy and Rachel Ward. Fast Cross-Polytope Locality-Sensitive Hashing. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 53:1-53:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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  author =	{Kennedy, Christopher and Ward, Rachel},
  title =	{{Fast Cross-Polytope Locality-Sensitive Hashing}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{53:1--53:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.53},
  URN =		{urn:nbn:de:0030-drops-81936},
  doi =		{10.4230/LIPIcs.ITCS.2017.53},
  annote =	{Keywords: Locality-sensitive hashing, Dimension reduction, Johnson-Lindenstrauss lemma}
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